Olivo-cerebellar controller

ABSTRACT

Non-linear control laws are disclosed and implemented with a controller and control system for maneuvering an underwater vehicle. The control laws change the phase of one Inferior-Olive (IO) neuron with respect to another IO. One control law is global, that is, the control law works (stable and convergent) for any initial condition. The remaining three control laws are local. The control laws are obtained by applying feedback linearization, while retaining non-linear characteristics. Each control law generates a profile (time history) of the control signal to produce a desired phase difference recognizable by a controller to respond to disturbances and to maneuver an underwater vehicle.

This application claims the benefit of U.S. Provisional PatentApplication Ser. No. 60/994,093, filed on Sep. 17, 2007 and which isentitled “Olivo-Cerebellar Controller” by the inventors, Sahjendra Singhand Promode R. Bandyopadhyay.

STATEMENT OF GOVERNMENT INTEREST

The invention described herein may be manufactured and used by or forthe Government of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefore.

CROSS REFERENCE TO OTHER PATENT APPLICATIONS

This application relates to U.S. patent application Ser. No. 11/901,546,filed on Sep. 14, 2007 and which is entitled “Auto-catalytic Oscillatorsfor Locomotion of Underwater Vehicles” by the inventors Promode R.Bandopadhyay, Alberto Menozzi, Daniel P. Thivierge, David Beal andAnuradha Annnaswamy.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

The present invention relates to a controller and control system for anunderwater vehicle; specifically, a controller and control system whichutilize non-linear dynamics supported by underlying mathematics tocontrol the propulsors of an underwater vehicle.

(2) Description of the Prior Art

Future underwater platforms are expected to have numerous sensors andperformance capabilities that will mimic the capabilities of aquaticanimals. A key component of such platforms would be their controller.Because such platforms, and an existing U.S. Navy Biorobotic AutonomousUndersea Vehicle (BAUV) is an example, keep station in highly disturbedfields near submarines or in the littoral areas, it is essential for theplatforms (vehicles) to have quick-responding controllers for theirpropulsor systems.

Hydrodynamic models based on conventional engineering controllers havenot been able to produce the desired levels of control. Thus, abiology-inspired controller is a realistic alternative. Because thebrains of animals perform complex tasks which rely on nonlineardynamics, the underlying mathematics provide a foundation for thecontroller and control system of the present disclosure.

Traditional control systems are designed using linear models obtained byJacobian linearization. This linearization allows design using frequencydomain techniques (such as lag-lead compensation, PID feedback, etc.)and a state-space approach (linear optimal control, pole assignment,servo-regulation, adaptive control, etc.). However, any controllerdesigned using linearized models of the system will fail to stabilizeunless the perturbations are small.

One must use nonlinear design techniques if the control system is tooperate in a larger region. For underwater vehicles, linear andnonlinear control systems based on pole placement, feedbacklinearization, sliding mode control, and adaptive control, etc. havebeen designed. However, in these designs, it is assumed that the vehicleis equipped with traditional control surfaces. As such, these vehicleshave limited maneuvering capability.

For large and agile maneuvers, traditional control surfaces areinadequate and new control surfaces must be developed. Observations ofmarine animals provide the potential of fish-like oscillating fins forthe propulsion and maneuvering of autonomous underwater vehicles (AUVs).AUVs exist with multiple oscillating fins which impart high lift andthrust. The oscillatory motion of the fins or propulsors is obtained byinferior olives which provide robust command signals to controllers andservomotors of the fins.

Inferior olives have complex nonlinear dynamics and have robust andunique self-oscillation [(Limit Cycle Oscillation (LCO)]characteristics. Efforts have been made to model the inferior olives(IO). Limited results on phase control of IOs in an open-loop sense areavailable using a pulse type stimulus. However, the required pulseheight of the input signal which depends on the state of the IOs at theswitching instant as well as the target relative phase between the IOshas not been derived. For the application of the IOs to the AUV,closed-loop control systems must be developed for the synchronizationand phase control of the IOs.

SUMMARY OF THE INVENTION

It is therefore a general purpose and primary object of the presentinvention to provide control laws for the synchronization and phaseangle control of multiple inferior olives (IO) used in a maneuveringcontroller or control system of an underwater vehicle;

It is a further object of the present invention to provide non-linearcontrol laws that the controller or control system can use to change aphase of one IO with respect to another IO; and

It is a still further object of the present invention to provide aglobal control law for a controller to use in maneuvering an underwatervehicle; and

It is a still further object of the present invention to provide a localcontrol law for a controller to use in maneuvering an underwatervehicle.

In order to attain the objects described, the present invention providesclosed-loop control of multiple inferior olives (IOs) for maneuvering aBiorobotic Autonomous Undersea Vehicle (BAUV). A model of an ith IO isdescribed where variables are associated with sub-threshold oscillationsand low threshold spiking. Higher threshold spiking is also described.

For the sake of simplicity, the synchronization of only two IOs isconsidered, but it is seen that the approach is extendable for thesynchronization of any number of IOs.

In optimizing the controller or control system for maneuvering, thestate vector for the ith IO is defined and a nonlinear vector functionand constant column vector are obtained. Synchronization is defined byfirst considering the synchronization of two IOs having arbitrary andpossibly large initial conditions. Note that if a delay time is zero,the IOs oscillate in synchronism with a relative phase zero. However, ifone sets the delay time, the IO₁ will oscillate lagging behind the IO₂with a relative phase angle. Although, the convergence of thesynchronization error has been required to be only asymptotic; forpractical purposes, it will be sufficient if one can design a controlsystem for the IO₁ which is sufficiently fast.

In the disclosure, four control systems are presented for thesynchronization of two IOs based on an input-output feedbacklinearization (nonlinear inversion) approach. For the purpose of thedesign of the controller or control system, output variables associatedwith the nonlinear system. It is shown that the choice of the outputvariable is important in shaping the behavior of the closed-loop system;although, by following the approach presented, various input-outputlinearizing control systems can be obtained.

The derivation of a control law is considered for the globalsynchronization of the IO₁ with the reference IO₂. It is desired todesign a synchronizing control system such that IO₁ oscillates insynchronism with a delay time corresponding to a desired phase anglewith respect to the reference IO₂. In global synchronization, thesynchronization is accomplished for all values of initial conditions ofthe two IOs. The output function is a function of the state vectors ofIO₁ and IO₂. This choice of the output yields the global result.

For the nonlinear closed-loop system, the output satisfies a fourthorder linear differential equation. One can choose larger gains toobtain faster convergence to zero. For the chosen output, because thesystem is of dimension four and the relative degree is four, thedimension of the zero dynamics is null. The zero dynamics represent theresidual dynamics of the system when the output error is constrained tobe zero.

The frequency of oscillation of the IOs depends on the systemparameters. Signals of different frequencies can be obtained by timescaling. In the disclosure it is observed that the IOs are not initiallyin phase. As the controller switches, the IOs synchronize. However, asthe command changes, it causes larger deviations in the tracking oftrajectories due to a large control input.

The controller or the control system uses feedback of nonlinearfunctions of state variables and has a global synchronization property.The complexity and performance of the controller depends on the choiceof the output function.

The IOs will synchronize if the equilibrium point is asymptoticallystable (globally asymptotically stable). For asymptotic analysis,ignoring a decaying part, which represents the deviation of a trajectoryfrom, a periodic signal can be represented by a Fourier series.Moreover, the amplitude of the harmonic converges to zero and forstability analysis a finite number of harmonics will suffice.

A simple control law has linear feedback terms involving only {tildeover (z)} and {tilde over (w)} variables and are independent of u_(i)and v_(i) variables. The output {tilde over (w)} satisfies a first-orderequation and in the closed-loop system {tilde over (w)} tends to zero.However, the stability in the closed-loop system will depend on thestability property of the zero dynamics. Apparently if the origin (ũ,{tilde over (v)}, {tilde over (z)})=0 of the zero dynamics isasymptotically stable, then {tilde over (x)} converges to zero as {tildeover (w)} tends to zero.

The relative merits of the four controllers are such that the firstcontroller has a global stabilization property and the remainingcontrollers have established local synchronization. It is expected thatas the complexity of control law increases, the region of stabilityenlarges. For this reason, one expects that the control law canaccomplish synchronization for relatively small perturbations at theinstant when the phase command is given. Of course, the error, andtherefore the synchronization of the IOs, depends on the instant ofcontroller switching. Based on simulation results, it has been foundthat two control laws for the controllers have fairly large regions ofstability and one control law does not necessarily have to use anothercontrol law.

Unlike the global control laws for the controller, the local controllaws provide smoother responses. This is due to a fast-varying nonlinearfunction of large magnitude in the control law. There exists flexibilityin the design, and by a proper choice of feedback gains and thereference phase command signals, one can obtain different responsecharacteristics. This flexibility in phase control of IOs is useful inperforming desirable maneuvers for the BAUV.

One must note that the profile of the control signal will depend on thestates of the IOs when a pulse is applied. The derived controllers arebased on the input-output feedback linearization theory, as well asstability and convergence. The control system can be switched on forphase control at any instant since the system utilizes state variablefeedback and one can command the IO to follow a sequence of phase changewhen needed.

BRIEF DESCRIPTION OF THE DRAWINGS

Further objects and advantages of the invention will become readilyapparent from the following detailed description and claims inconjunction with the accompanying drawings wherein;

FIG. 1A-1D are each a graph depicting global synchronization usingcontrol law C_(u) with IO₁ commanded to track IO₂ with a delay 0.125 fort ε [0, 4), 0.25 for t ε [4, 6), 0.5 for t ε [6, 8) and 0.75 for t ε [8,10) with the controller of IO₁, switching at two seconds, plots are ofx₁(t) and x₂(t−t_(d));

FIG. 2A-2D are each a graph depicting global synchronization usingcontrol law, C_(u), plots are of x₂(t) and x₂(t−t_(d)) for commandinputs of FIG. 1A-1D;

FIG. 3A-3D are each a graph depicting global synchronization usingcontrol law, C_(u), plots are of u_(i) (t), v_(i)(t), z_(i) (t) andcontrol inputs I_(est1), I_(ext2) for command inputs of FIG. 1A-1D;

FIG. 4A-4D are each a graph depicting local synchronization usingcontrol law C_(v) with IO₁ commanded to track IO₂ with a delay 0.125 fort ε [0, 4), 0.25 for t ε [4, 6), 0.5 for t ε [6, 8) and 0.75 for t ε [8,10) where the controller of IO₁ switches at two seconds, plots are ofx₁(t) and x₂(t−t_(d));

FIG. 5A-5D are each a graph depicting local synchronization usingcontrol law, C_(v), plots of u₁ (t), v₁ (t), z₁ (t) and control inputsI_(ext1), I_(ext2) for the command inputs of FIG. 1A-FIG. 1D;

FIG. 6A-6D are each a graph depicting local synchronization usingcontrol law C_(z) with IO₁ commanded to track IO₂ with a delay 0.125 fort ε [0, 4), 0.25 for t ε [4, 6), 0.5 for t ε [6, 8), and 0.75 for t ε[8, 10) with the controller of IO₁ switching at two seconds, plots areof x₁(t) and x₂(t−t_(d));

FIG. 7A-7D are each a graph depicting local synchronization usingcontrol law, C_(z), plots of u₁ (t), v₁ (t), z₁ (t) and control inputsI_(ext1), I_(ext2) for the command inputs of FIG. 1A-1D;

FIG. 8A-8D are each a graph depicting local synchronization usingcontrol law C_(w) with IO₁ commanded to track IO₂ with a delay 0.125 fort ε [0, 4), 0.25 for t ε [4, 6), t ε [6, 8) and 0.75 for t ε [8, 10)with the controller IO₁ switching at two seconds, plots are of x₁(t) andx₂(t−t_(d));

FIG. 9A-9D are each a graph depicting local synchronization usingcontrol law, C_(w), plots of u₁(t), v₁(t), z₁(t) and control inputsI_(ext1), I_(ext2) for the command inputs of FIG. 1A-1D;

FIG. 10A-10D are each a graph depicting local synchronization usingcontrol law C_(w) (faster oscillation) with IO₁ commanded to track IO₂with a delay 0.125 for t ε [0, 4), 0.25 for t ε [4, 6], 0.5 for t ε [6,8) and 0.75 for t ε [8, 10) with the controller IO₁ switching at twoseconds, plots are of x₁(t) and x₂(t−t_(d));

FIG. 11A-11D are each a graph depicting synchronization, plots are ofx₂(t) and x₂(t−t_(d)) for the command inputs of FIG. 1A-1D; and

FIG. 12A-12D are each a graph depicting local synchronization usingcontrol law C_(W) (faster oscillation), plots are of u₁ (t) , v₁ (t), z₁(t) and control inputs I_(ext1), I_(ext2) and control inputs for thecommand inputs of FIG. 1A-1D.

DETAILED DESCRIPTION OF THE INVENTION

Referring now to the present disclosure, a subsection on inferior-olivesand a practical application of control laws affecting inferior-olivesare presented.

Inferior Olives Model and Synchronization

This disclosure focuses on closed-loop control of multiple inferiorolives (IOs) for maneuvering Biorobotic Autonomous Undersea Vehicles(BAUVs). The model of an ith IO is described by

$\begin{matrix}{\begin{bmatrix}{\overset{.}{u}}_{i} \\{\overset{.}{v}}_{i} \\{\overset{.}{z}}_{i} \\{\overset{.}{w}}_{i}\end{bmatrix} = {\begin{bmatrix}{k \in_{Na}^{- 1}\left( {{p_{iu}\left( u_{i} \right)} - v_{i}} \right)} \\{k\left( {u_{i} - z_{i} + I_{Ca} - I_{Na}} \right)} \\{{p_{iz}\left( z_{i} \right)} - w_{i}} \\{\in_{Ca}\left( {z_{i} - I_{Ca}} \right)}\end{bmatrix} + {\begin{bmatrix}0 \\0 \\0 \\{- \in_{Ca}}\end{bmatrix}\mspace{14mu}{I_{exit}(t)}}}} & (1)\end{matrix}$where the variables “z_(i)” and “w”, are associated with thesub-threshold oscillations and low threshold (Ca-dependent) spiking, and“u_(i)” and “v_(i)” describe the higher threshold (Na⁺-dependent)spiking. The constant parameters ε_(Ca) and ε_(Na) control theoscillation time scale; I_(ca) and I_(Na) drive the depolarizationlevels; and k sets a relative time scale between the uv- andzw-subsystems.

The nonlinear functions are:p _(iu)(u _(i))=u _(i)(u _(i) −a)(1−u _(i))p _(iz)(z _(i))=z _(i)(z _(i) −a)(1−z _(i))   (2)“p” being a non-linear function and “a” is a constant parameter.

The function I_(exti) (t) is the extra-cellular stimulus which is usedhere for the purpose of control.

Definex _(i)=(u _(i) ,v _(i) ,z _(i) ,w _(i))^(T) εR ⁴   (3)where “x” is the state vector of the ith IO, “R” is the set of realnumbers. Equation (1) can be written in a compact form as{dot over (x)} _(i) =f _(i)(x _(i))+g _(i) u _(ci)   (4)where u_(ci)=I_(exti) is the control input of the ith IO and “f”, “g”are vectors. The nonlinear vector function f_(i) (x_(i))εR⁴ and theconstant column vector g_(i) are obtained from Equation (1). It is knownto those skilled in the art that a system utilizing Equation (1)exhibits limit cycle oscillations. Using harmonic balancing, it ispossible to predict the approximate magnitudes, frequency and phases ofperiodic solutions of the components of the system.

As stated, the primary objective is to develop control laws for thesynchronization and phase angle control of multiple IOs for t he purposeof BAUV control. For the sake of simplicity, the synchronization of onlytwo IOs is considered, but it is seen that the approach is extendablefor the synchronization of any number of IOs. Synchronization is definedfirst.

Consider two IOs{dot over (x)} ₁ =f ₁(x ₁)+g ₁ u _(c1){dot over (x)} ₂ =f ₂(x ₂)+g ₂ u _(c2).   (5)

Suppose that the state vector x₂ of the second IO is treated as thereference signal.

Consider a solution x₂(t) of the IO₂ beginning from an initial conditionx₂₀, with an input u_(c2)=0 set to zero and let x₂(t−t_(d)) [“t” beingtime] be the delayed signal obtained from x₂(t), where t_(d)>0 is anarbitrary delay time. Then for the prescribed delay time t_(d), IO₁ issaid to be asymptotically synchronized to the IO₂ if the error signal{tilde over (x)}(t)=x₁(t)−x₂(t−t_(d)) converges to zero as t tends to ∞[infinity].

Consider the synchronization of the two IOs having arbitrary andpossibly large initial conditions. Note that if the delay time is zero,x₁(t)−x₂(t) diminishes to zero as time progresses and the IOs oscillatein synchronism with a relative phase zero. However, if one sets thedelay time as t_(d)=φ/(2πf) (“f” is the period of oscillation of theIO₂), the IO₁ will oscillate lagging behind the IO₂ with a relativephase angle φ. Although, the convergence of the synchronization error,has been required to be only asymptotic, for practical purposes, it willbe sufficient if one can design the control system for the IO₁ which issufficiently fast.

Synchronizing Control Systems

Four control systems are presented for the synchronization of the twoIOs based on an input-output feedback linearization (nonlinearinversion) approach. For the purpose of the design, consider outputvariables associated with the nonlinear system for Equation (5) of theforme=h(x ₁(t), x ₂(t−t _(d))).   (6)

Later “h”, which is a function of the state variables of the two IOs, isselected to meet the desired objective. It will be seen that the choiceof the output variable “e” is important in shaping the behavior of theclosed-loop system. Although, by following the approach presented here,various input-output linearizing control systems can be obtained,derivation of the four control systems of varying complexity andsynchronizing characteristics are considered.

Global Synchronization: Control Law (C_(u))

Now consider the derivation of a control law for the globalsynchronization of the IO₁ with the reference IO₂. The reference IO₂ hasan input I_(ext2)=0. It is desired to design a synchronizing controlsystem such that IO₁ oscillates in synchronism with a delay time oft_(d) seconds corresponding to a desired phase angle φ with respect toreference IO₂. By global synchronization, the synchronization must beaccomplished for all values of initial conditions x _(iO) ε R⁴, i=1,2 ofthe two IOs.

For the purpose of design, the controlled output variable is chosen as:e(t)=h _(u)(x ₁(t), x ₂(t−t _(d)))=u ₁(t)−u ₂(t−t _(d)).   (7)

Note that the output function “e” is a function of only the firstcomponent of the state vectors of IO₁ and IO₂ at time t and t−t_(d),respectively. But it will be seen later that this choice of the output“e” yields the global result. The subscript “u” of the function “h”denotes dependence on the variables “u_(i)”.

For compactness, define the composite state vector for the two IOs asx_(a)(t)=(x₁(t)^(T), x₂(t−t_(d))^(T) ε R⁸, where “T” denotes matrixtransposition. Then from Equation (5), one has

$\begin{matrix}{{{\overset{.}{x}}_{a}(t)} = {\quad{\begin{bmatrix}{{\overset{.}{x}}_{1}(t)} \\{{\overset{.}{x}}_{2}\left( {t - t_{d}} \right)}\end{bmatrix} = {{\begin{bmatrix}{f_{1}\left( {x_{1}(t)} \right)} \\{f\left( {x_{2}\left( {t - t_{d}} \right)} \right)}\end{bmatrix} + {\begin{bmatrix}g_{1} \\0\end{bmatrix}\mspace{14mu}{u_{c\; 1}(t)}}}\overset{.}{=}{{f\left( {x_{a}(t)} \right)} + {{{gu}_{c\; 1}(t)}.}}}}}} & (8)\end{matrix}$

The state error ({tilde over (x)}=x₁(t)−x₂(t−t_(d))) dynamics and theassociated output e can be written as

$\begin{matrix}{{\overset{.}{\overset{\sim}{x}} = {{{f_{1}\left( {{\overset{\sim}{x}(t)} + {x_{2}\left( {t - t_{d}} \right)}} \right)} - {f_{2}\left( {x_{2}\left( {t - t_{d}} \right)} \right)} + {g_{1}{u_{c\; 1}(t)}}}\overset{.}{=}{{f_{e}\left( {{\overset{\sim}{x}(t)},t} \right)} + {g_{1}{u_{c\; 1}(t)}}}}}{e = {{h_{u}\left( {x_{a}(t)} \right)} = {h_{u}\left( {\overset{\sim}{x}(t)} \right)}}}} & (9)\end{matrix}$where f_(e)({tilde over (x)},t)=f₁({tilde over(x)}(t)+x₂(t−t_(d)))−f₂(x₂(t−t_(d))) is defined. Note that argument “t”has been used in “f_(e)” to indicate dependence on the bounded and knowndelayed reference state vector of the unforced IO₂. Thus, the system ofEquation (9) can be treated as a nonautonomous system of dimension four.

Define the Lie derivative of the function h_(u) along the vector field fas

$\begin{matrix}{{{L_{f}{h_{u}\left( {x_{a}(t)} \right)}} = {{\frac{\partial h}{\partial x_{a}}{f\left( {x_{a}(t)} \right)}} = {{\frac{\partial h_{u}}{\partial x_{1}}{f_{1}\left( {x_{1}(t)} \right)}} + {\frac{\partial h_{u}}{\partial x_{2}}{f_{2}\left( {x_{2}\left( {t - t_{d}} \right)} \right)}}}}}{{{{and}\mspace{14mu}{for}\mspace{14mu} k} = 0},1,{{2\mspace{11mu}\ldots}\mspace{11mu};{{{and}\mspace{14mu}{let}\mspace{14mu} L_{f}^{j}{h_{u}\left( x_{a} \right)}} = {L_{f}{L_{f}^{j}\left( {x_{a}(t)} \right)}\mspace{14mu}{and}}}}}} & (10) \\{{L_{g}L_{f}^{k}{h_{u}\left( x_{a} \right)}} = {\frac{{\partial L_{f}^{k}}h_{u}}{\partial x_{a}}{g.}}} & (11)\end{matrix}$

For the system of Equation (8), computing the Lie derivatives, it isverified that for j=0,1,2,3, one has

$\begin{matrix}{{{e^{(j)}(t)} = {L_{f}^{j}{h_{u}\left( {x_{a}(t)} \right)}}}{{{and}\mspace{14mu}{for}\mspace{14mu} j} = {4\mspace{14mu}{gives}}}} & (12) \\{{e^{(j)}(t)} = {{{L_{f}^{j}{h\left( {x_{a}(t)} \right)}} + {L_{g}L_{f}^{j - 1}{h\left( {x_{a}(t)} \right)}{u_{c\; 1}(t)}}}\overset{.}{=}{{a_{u\; 1}\left( {\overset{\sim}{x},t} \right)} + {b_{u\; 1}u_{c\; 1}}}}} & (13)\end{matrix}$where e^((k))=de^(k)/dt^(k) and one can show that b_(u1)=k²ε_(Ca)/ε_(Na). For the nonautonomous system of Equation (9), defining

$\begin{matrix}{{L_{fe}\left( . \right)} = {\frac{\partial\left( . \right)}{\partial t} + {\frac{\partial\left( . \right)}{\partial\overset{\sim}{x}}{f_{e}\left( {t,\overset{\sim}{x}} \right)}}}} & (14) \\{{{{L_{fe}^{j}{h_{u}\left( {\overset{\sim}{x},t} \right)}} = {L_{f}^{j}{h_{u}\left( x_{a} \right)}}},{j = 0},1,\ldots\mspace{11mu},4}{{{and}\mspace{14mu} L_{g\; 1}L_{fe}^{3}{h_{u}\left( {\overset{\sim}{x},t} \right)}} = {b_{u\; 1}.}}} & (15)\end{matrix}$Since the control input appears in the fourth derivative of the output efor the first time for the system utilizing Equation (9), the output eis of the relative degree r=4.

In view of Equation (13), an input-output linearizing control law isselected as

$\begin{matrix}{{\overset{.}{u}}_{c\; 1} = {{b_{u\; 1}^{- 1}\text{(}} - a_{u\; 1} - {\sum\limits_{j = 0}^{3}{p_{j}L_{f}^{j}{h_{u}\left( {x_{a}(t)} \right)}}}}} & (16)\end{matrix}$where p_(j), j=0,1,3, are the constant feedback gains and “b” is avector. Because e^((j))(t)=L_(f) ^(j)h_(u)(x_(a)(t)), substituting thecontrol law of Equation (16) in Equation (13) gives an output equationof the forme ⁽⁴⁾ +p ₃ e ⁽³⁾ +p ₂ e ⁽²⁾ +p ₁ ė+p ₀ e=0   (17)

For the nonlinear closed-loop system of Equations (9) and (16), theoutput e(t) satisfies a fourth order linear differential equation. Thegains p_(j) are chosen such that Equation (17) is exponentially stable,and thereby e(t) and derivatives of e(t) converge to zero as t tends toinfinity. Of course, one can choose larger gains to obtain fasterconvergence of e(t) to zero. For the chosen output, because the systemof Equation (9) is of dimension four and the relative degree of e isfour, the dimension of the zero dynamics is null. The zero dynamicsrepresent the residual dynamics of the system when the output error e(t)is constrained to be zero.

In fact, there exists a diffeomorphism P_(u) for t ε[0,∞) mapping R⁴into R⁴ such that {tilde over (x)}=P_(u)(ξ,t), where ξ=(e, ė, ë,e⁽³⁾)^(T) ε R⁴. One can find the map P_(u). First of all, one has ũ=e,where {tilde over (x)}=x₁(t)−x₂(t−t_(d))=(ũ,{tilde over (v)},{tilde over(z)},{tilde over (w)})^(T) is defined. Using Equation (12) one can showthat

$\begin{matrix}{\overset{\sim}{x} = {{P_{u}\left( {\xi,t} \right)} = \begin{bmatrix}e \\{q_{1}\left( {e,\overset{.}{e},t} \right)} \\{q_{2}\left( {e,\overset{.}{e},\overset{¨}{e},t} \right)} \\{q_{3}\left( {e,\overset{.}{e},\overset{¨}{e},e^{(3)},t} \right)}\end{bmatrix}}} & (18)\end{matrix}$whereq ₁=−ε_(Na) k ⁻¹ ė+p _(1u)(e+u ₂(t−t _(d)))−p _(2u)(u ₂(t−t _(d))), q ₂=−{dot over (q)} ₁ k ⁻¹ +e, and q ₃ =−{dot over (q)} ₂ +p _(1z)({tildeover (z)}+z ₂(t−t _(d)))−p _(2z)(z ₂(t−t _(d))).Note that the argument “t” in “q_(i)” and “P_(u)” indicates dependenceon the reference trajectory x₂(t−t_(d)) and derivatives of the referencetrajectory. Furthermore, it can be verified that Pu (0, t)=0; that is,{tilde over (x)}=0 when e and derivatives of e vanish. Because P_(u) isa diffeomorphism, P_(u)(0, t)=0, and the linear system of Equation (17)is exponentially stable, global synchronization of the IOs isaccomplished and the two IOs oscillate together but with the requiredrelative phase. Note that the control stimulus, I_(ext1), vanishes whenthe IOs capture the unique limit cycle; only the IO₁ falls behind by thedelay time t_(d) (phase angle φ).

To examine the synchronizing capability of the control system, theclosed-loop system including the IOs given in Equation (5) and thecontrol law of Equation (16) is simulated. The parameters of the IOsselected are: E_(Na)=0.001, E_(Ca)=0.02, k=0.1, I_(Ca)=0.018,I_(Na)=−0.61, and a=0.015. One can use another set of parameters aswell. The input to IO₂ is kept to zero. The feedback gains chosen aresuch that the poles of Equation (17) are at 25(−0.424±j 1.263) and25(−6.26±j 0.4141). These poles have been selected to obtain goodtransient responses by observing the simulated responses, however onecould choose other pole locations as well for synchronization. Theinitial conditions are x₁₀=(0.4, 0.6, 0.4, 0.5)^(T) and x₂₀=(0.2, 0.4,0.2, 0.3)^(T). Thus the initial condition of the IOs differs. Thefrequency of oscillation of the IOs depends on the system parameters.Signals of different frequencies can be obtained by time scaling. Forillustration, a time scaling is introduced by multiplying thederivatives of the variables by a scaling factor of sixty.

It is desired to have the delay time t_(d) as 0.125 for t ε [0,4), 0.25for t ε [4, 6), 0.5 for t ε [6, 8) and 0.75 for t ε [8, 10),respectively. The controller is switched on at t=2 (sec), that isI_(ext1)=0 for t<2 and the delay command changes every two seconds.Referring now to the drawings, responses are shown in FIG. 1( a)-(d),FIG. 2( a)-(d) and FIG. 3( a)-(d). In the figures, the variables with asubscript “d” indicate delayed values (such as u2_(d) denotingu₂(t−t_(d))). It is observed that the IOs are not initially in phase. Asthe controller switches at two seconds, the IOs synchronize having adelay time of 0.125 seconds. The command changes at four, six, and eightseconds to delay times of 0.25, 0.5 and 0.75 seconds. Following eachcommand, x₁(t) tracks x₂(t−t_(d)) and it is seen that u₁(t)−u₂(t−t_(d))and v₁(t)−v₂(t−t_(d)) remain close to zero after two seconds. However,as the command changes, it causes larger deviations in the tracking ofz- and w-trajectories due to large control input acting on the system.Note that a comparatively large spike appears in the control input attwo seconds and subsequently smaller magnitudes of control input arerequired each time that the command changes. Simulation has been donefor other initial conditions and a parameter value of a. It is foundthat frequency changes with a, but for a low value of a=0.01, u-responsehas a sharper spike.

The controller C_(u) uses feedback of nonlinear functions of the statevariables and has a global synchronization property. A controller usingfewer state components and/or nonlinear feedback functions will benotable for implementation. The complexity and performance of thecontroller depends on the choice of the output function e. The existenceof simpler controllers using different controlled output variables isexamined in the next subsections.

Local Synchronization: Control Law (C_(v))

Now consider the derivation of a control law (termed as C_(v)) for thechoice of controlled output variablee(t)=h _(v)(x _(a)(t))=v₁(t)−v₂(t−t _(d))={tilde over (v)}(t).   (19)Note that the same symbol “e” is used to indicate a different function.For this choice of e, that for j=0, 1, 2, one has

$\begin{matrix}{{e^{(j)}(t)} = {{L_{f}^{j}{h_{v}\left( {x_{a}(t)} \right)}\mspace{14mu}{and}\mspace{14mu}{for}\mspace{14mu} j} = {3\mspace{14mu}{gives}}}} & (20) \\{{e^{(j)}(t)} = {{{L_{f}^{j}{h_{v}\left( {x_{a}(t)} \right)}} + {L_{g}L_{f}^{j - 1}{h_{v}\left( {x_{a}(t)} \right)}{u_{c\; 1}(t)}}}\overset{.}{=}{{a_{v\; 1}\left( {\overset{\sim}{x},t} \right)} + {b_{v\; 1}u_{c\; 1}}}}} & (21)\end{matrix}$where one can show that b_(v1)=−k ε_(Ca). Since the control inputappears in the third derivative of the output e for the first time forthe system of Equation (9), the output e has the relative degree r=3.

In view of Equation (21), an input-output linearizing control law isselected as

$\begin{matrix}{u_{c\; 1} = {b_{v\; 1}^{- 1}\left( {{- a_{v\; 1}} - {\sum\limits_{j = 0}^{2}{p_{j}L_{f}^{j}{h_{v}\left( {x_{a}(t)} \right)}}}} \right)}} & (22)\end{matrix}$where p_(j), j=0, 1, 2, are the constant feedback gains. Substitutingthe control law of Equation (22) in Equation (21) gives the outputequation of the forme ⁽³⁾ +p ₂ e ^((2)i +p) ₁ ė+p ₀ e=0.   (23)

The gains p₁ are chosen such that the characteristic polynomialΠ_(v)(λ)=λ³ +p ₂λ² +p ₁ λ+p ₀.   (24)associated with Equation (23) is Hurwitz, commonly known in the art.Hurwitz means that the roots of Π_(v)(λ)=0 have real part negative. Forthe choice of such parameters, e and the derivatives tend to zero.

For the nonlinear closed-loop system of Equation (9) and Equation (22),the output e(t) satisfies a third-order linear differential equation.Because the system of Equation (9) is of dimension four and the relativedegree or e is three, the dimension of the zero dynamics is one. Infact, there exists a diffeomorphism P_(v) for tε[0,∞) mapping R⁴ into R⁴such that {tilde over (x)}=P_(v)(ξ,t) where ξ is now defined asξ=(ũ,e,ė,ë)^(T). Using Equation (20) one can show that

$\begin{matrix}{\mspace{79mu}{{\overset{\sim}{x} = {{P_{v}\left( {\xi,t} \right)} = \begin{bmatrix}\overset{\sim}{u} \\e \\{\overset{\sim}{u} - {k^{- 1}\overset{.}{e}}} \\{q_{v}\left( {\overset{\sim}{u},e,\overset{.}{e},\overset{¨}{e},t} \right)}\end{bmatrix}}}\mspace{79mu}{where}}} & (25) \\{q_{v} = {k \in_{Na}^{- 1}{\left( {{- {p_{1u}\left( {\overset{\sim}{u} + {u_{2}\left( {t - t_{d}} \right)}} \right)}} + {p_{2u}\left( {u_{2}\left( {t - t_{d}} \right)} \right)} + e} \right) + {p_{1z}\left( {\overset{\sim}{z} + {z_{2}\left( {t - t_{d}} \right)}} \right)} - {p_{2z}\left( {z_{2}\left( {t - t_{d}} \right)} \right)} + {k^{- 1}\overset{¨}{e}}}}} & (26)\end{matrix}$and it is understood that {tilde over (z)} is replaced by ũ−ė/k inq_(v). Furthermore, it can be verified that P_(v)(0, t)=0. However, theconvergence of the error “e” and the derivative to zero does notnecessarily imply the convergence of {tilde over (x)} to the origin. Forthe synchronization of the IOs, the stability property of the residualdynamics (the zero dynamics) must be examined when e vanishes.

It can be shown that the zero dynamics (when e=0) is given by

$\begin{matrix}{\overset{.}{\overset{\sim}{u}} = {{{- {ka}} \in_{Na}^{- 1}{\overset{\sim}{u} + k} \in_{Na}^{- 1}\left\lbrack {{\left( {1 + a - {3{u_{2}\left( {t - t_{d}} \right)}}} \right){\overset{\sim}{u}}^{2}} + {\left( {{2\left( {1 + a} \right){u_{2}\left( {t - t_{d}} \right)}} - {3{u_{2}^{2}\left( {t - t_{d}} \right)}}} \right)\overset{\sim}{u}} - {\overset{\sim}{u}}^{3}} \right\rbrack}\overset{.}{=}{g_{c}\left( {\overset{\sim}{u},{u_{2}\left( {t - t_{d}} \right)}} \right)}}} & (27)\end{matrix}$

The IOs will synchronize in a local (global) sense only if theequilibrium point ũ=0 is asymptotically stable (globally asymptoticallystable). The system of Equation (27) is a nonlinear nonautonomous systemand depends on the state u₂(t−t_(d)) of the reference IO. It is seenthat the solution of Equation (27) is bounded, because for large ũ,g_(c) is dominated by −ũ³.

For the stability analysis, consider the solutions of the zero dynamicsin a sufficiently small open set Ω_(u) around ũ=0. If u₂(t−t_(d)) issufficiently small, one has (∂g_(c)(0,t)/∂ũ)<0, and therefore ũ=0 of thezero dynamics is exponentially stable and the controller accomplisheslocal synchronization.

Alternatively, one can establish asymptotic stability of the zerodynamics using a center manifold theorem known to those ordinarilyskilled in the art. First note that, the solution x₂(t−t_(d)) of thereference IO converges to a closed orbit Γ₂. For asymptotic analysis,ignoring the decaying part, which represents the deviation of thetrajectory from Γ₂, the periodic signal u₂(t−t_(d)) can be representedby a Fourier series. Moreover, the amplitude of the kth harmonicconverges to zero as k tends to infinity and for stability analysis afinite number (N, a sufficiently large integer) of harmonics willsuffice. Let ω_(e) be the fundamental frequency of oscillation of thereference IO. As such, in the steady-state, it can be assumed thatu₂(t−t_(d)) can be generated by an exosystem{dot over (x)}_(e)=Λx_(e)   (28)and u₂(t−t_(d))=C₀x_(e) for row vector C_(O), where the block diagonalmatrix Λ is

$\begin{matrix}{\Lambda = {{diag}{\left\{ {0,\begin{bmatrix}0 & {{- n}\;\omega_{e}} \\{n\;\omega_{e}} & 0\end{bmatrix},{n = 0},1,2,{\ldots\mspace{11mu} N}} \right\}.}}} & (29)\end{matrix}$Assume that x_(e) ε Ω_(xe) and that the set Ω_(xe) is sufficientlysmall. This implies that u₂(t−t_(d)) is small. Since Equation (27) is afunction of x_(e) and Equation (27) is stable, there exists an invariantmanifold ũ(t)=Ũ(x_(e)) which satisfies the partial differential equation

$\begin{matrix}{{\frac{\partial\overset{\sim}{U}}{\partial x_{e}}\Lambda\; x_{e}} = {{g_{c}\left( {{\overset{\sim}{U}\left( x_{e} \right)},x_{e}} \right)}.}} & (30)\end{matrix}$

In view of the form of the function g_(c)(ũ,u₂(t−t_(d))), Equation (30)has a trivial solution Ũ=0, and moreover for small initial conditionsũ(0), the solution of Equation (27) satisfies∥ũ(t)−Ũ∥≦δ _(e) ^(−μt) ∥ũ(0)−Ũ∥  (31)where “δ” and “μ” are positive numbers. Since Ũ=0, according to Equation(31), it follows that for small u (t−t_(d)), ũ converges exponentiallyto zero and this establishes local synchronization of the IOs becauseP_(v) is diffeomorphic. However, only local synchronization of the IOsis established using the control law of Equation (22).

The closed-loop system including the control law of Equation (22) issimulated. The initial conditions, phase command signals and the modelparameters of FIG. 1(A)-(D) are retained. The feedback parameters p_(i)now correspond to the poles −3.5405 and −5(0.521±j1.0681) of thepolynomial Π_(v)(λ). Simulated responses are shown in FIG. 4(A)-(D) andFIG. 5(A)-(D). Observe that the IOs synchronize following each phasecommand. The control magnitude is smaller [see FIG. 3(A)-(D)] since thegains chosen are relatively small in this case. Although, it is not easyto establish global stability, it has been found by simulation thatsynchronization is accomplished for larger values of the initialconditions and different phase command sequences.

Local Synchronization: Control Law (C_(z))

Consider the derivation of a control law based one(t)=z ₁(t)−z ₂(t−t _(d))=h _(z)(x _(a))   (32)as the controlled output. For this choice of “e” it is easily verifiedthat for j=0,1, one has

$\begin{matrix}{{{e^{(j)}(t)} = {L_{f}^{i}{h_{z}\left( {x_{a}(t)} \right)}}}{{{and}\mspace{14mu}{for}\mspace{14mu} j} = {2\mspace{14mu}{gives}}}} & (33) \\{{e^{(j)}(t)} = {{{L_{f}^{i}{h_{z}\left( {x_{a}(t)} \right)}} + {L_{g}L_{f}^{j - 1}{h_{z}\left( {x_{a}(t)} \right)}u_{c\; 1}}}\overset{.}{=}{{a_{z\; 1}\left( {\overset{\sim}{x},t} \right)} + {b_{z\; 1}u_{c\; 1}}}}} & (34)\end{matrix}$where one can show that b_(z1)=ε_(Ca). Since the control input appearsin the second derivative of the output e for the first time for thesystem of Equation (9), the output e has the relative degree r=2.

In view of Equation (34), an input-output linearizing control law isselected as

$\begin{matrix}{u_{c\; 1} = {{b_{z\; 1}^{- 1}\text{(}} - a_{z\; 1} - {\sum\limits_{j = 0}^{1}{p_{j}L_{f}^{j}{h_{v}\left( {x_{a}(t)} \right)}}}}} & (35)\end{matrix}$where p_(j), j=0,1, are the constant feedback gains. Substituting thecontrol law of Equation (35) in Equation (34) gives the output equationof the forme ⁽²⁾ +p ₁ ė+p ₀ e=0.   (36)

The gains p_(i) are chosen such that the characteristic polynomialΠ_(z)(λ)=λ² +p ₁ λ+p ₀   (37)associated with Equation (36) is Hurwitz.

The zero dynamics in this case are described by the Equations

$\begin{matrix}{\mspace{79mu}{{\begin{bmatrix}\overset{.}{\overset{\sim}{u}} \\\overset{.}{\overset{\sim}{v}}\end{bmatrix} = {{\begin{bmatrix}{{- {ak}} \in_{Na}^{- 1}} & {{- k} \in_{Na}^{- 1}} \\k & 0\end{bmatrix}\begin{bmatrix}\overset{\sim}{u} \\\overset{\sim}{v}\end{bmatrix}} + \begin{bmatrix}g_{u} \\0\end{bmatrix}}}\mspace{79mu}{where}}} & (38) \\{g_{u} = {k \in_{Na}^{- 1}{\quad\left\lbrack {{\left( {1 + a - {3{u_{2}\left( {t - t_{d}} \right)}}} \right){\overset{\sim}{u}}^{2}} + {\left( {{2\left( {1 + a} \right){u_{2}\left( {t - t_{d}} \right)}} - {3{u_{2}^{2}\left( {t - t_{d}} \right)}}} \right)\overset{\sim}{u}} - {\overset{\sim}{u}}^{3}} \right\rbrack}}} & (39)\end{matrix}$and a diffeomorphism p_(z)(ξ, t) exists such that {tilde over(x)}=P_(z)(ξ,t) where now ξ=(ũ,{tilde over (v)},e,ė)^(T), and

$\begin{matrix}{\overset{\sim}{x} = {{P_{z}\left( {\xi,t} \right)} = \begin{bmatrix}\overset{\sim}{u} \\\overset{\sim}{v} \\e \\{{- \overset{.}{e}} + {p_{1z}\left( {e + {z_{2}\left( {t - t_{d}} \right)}} \right)} - {p_{2z}\left( {z_{2}\left( {t - t_{d}} \right)} \right)}}\end{bmatrix}}} & (40)\end{matrix}$

It follows that if the origin (ũ,{tilde over (v)})=0 of the zerodynamics is asymptotically stable and (e,ė)→0, then ξ tends to zerowhich implies the convergence of {tilde over (x)} to zero.

For the parameters of the IO, the matrix

$\begin{matrix}{A_{z} = \begin{bmatrix}{{- {ak}} \in_{Na}^{- 1}} & {{- k} \in_{Na}^{- 1}} \\k & 0\end{bmatrix}} & (41)\end{matrix}$is Hurwitz (i.e., the eigenvalues have a negative real part). In thesteady state, g_(u) is a function of x_(e), the state of the exosystemof Equation (28). In this case, in view of the center manifold theorem,for x_(e) ε Ω_(xe), there exists an invariant manifold (ũ, {tilde over(v)})=(Ũ(x_(e)), {tilde over (V)}(x_(e))) which satisfies the set ofpartial differential equations

$\begin{matrix}{{{\frac{\partial\overset{\sim}{U}}{\partial x_{e}}\Lambda\; x_{e}} = {{- {ak}} \in_{Na}^{- 1}{{\overset{\sim}{U}\left( x_{e} \right)} - k} \in_{Na}^{- 1}{{\overset{\sim}{V}\left( x_{e} \right)} + {g_{u}\left( {{\overset{\sim}{U}\left( x_{e} \right)},x_{e}} \right)}}}}{{\frac{\partial\overset{\sim}{V}}{\partial x_{e}}\Lambda\; x_{e}} = {k\;{\overset{\sim}{U}\left( x_{e} \right)}}}} & (42)\end{matrix}$

These equations are satisfied by (Ũ(x_(e)), {tilde over (V)}(x_(e)))=0.

Similar to the arguments based on either the Jacobian linearization orthe center manifold theorem, it can be concluded that for smallu₂(t−t_(d)), the origin of the zero dynamics is exponentially stable (ina local sense), and thereby local synchronization is accomplished. Notethat this control law is simpler that C_(v).

Simulation results are now presented for the closed-loop system ofEquations (5) and (35). The parameter values, command input sequence,and the initial conditions of FIG. 1(A)-(D) are retained. The feedbackgains are chosen are so that the poles of the e-dynamics are at(−7.07±j7.072). Simulated responses are shown in FIG. 6(A)-(D) and FIG.7(A)-(D). Synchronization is accomplished and the (z and w)-responsesare smoother and control input is smaller than those obtained using thecontrol laws, C_(u) and C_(v). However, sharper peaking of u- andw-response is observable at certain instances, when the phase commandchanges. However, the stability results have been established only forthe local synchronization.

Local Synchronization: Control Law (C_(w))

A still simpler control law for the choice of the controlled outputvariable is:e(t)=w ₁(t)−w ₂(t−t _(d))={tilde over (w)}=h _(w)(x _(a)(t)).   (43)

For this choice, one hasė(t)=L _(f) h _(w)(x _(a)(t))+L _(g) h _(w)(x _(a)(t))u _(c1)(t)   (44)and the control law isu _(c1) ={tilde over (z)}(t)+p ₀ ε_(Ca) ⁻¹ {tilde over (w)}  (45)where p_(o) is any positive number. Thus the control law has simplelinear feedback terms involving only the {tilde over (z)} and {tildeover (w)} variables and are independent of u_(i) and v_(i).

The output {tilde over (w)} now satisfies a first-order equation{tilde over ({dot over (w)}+p ₀ {tilde over (w)}=0   (46)and in the closed-loop system {tilde over (w)} tends to zero. However,the stability in the closed-loop system will depend on the stabilityproperty of the zero dynamics which is now of dimension three.

The zero dynamics in this case are obtained by setting {tilde over(w)}=0 and can be shown to be described by

$\begin{matrix}{{\begin{bmatrix}\overset{.}{\overset{\sim}{u}} \\\overset{.}{\overset{\sim}{v}} \\\overset{\sim}{z}\end{bmatrix} = {{{\begin{bmatrix}{{- {ak}} \in_{Na}^{- 1}} & {{- k} \in_{Na}^{- 1}} & 0 \\k & 0 & {- k} \\0 & 0 & {- a}\end{bmatrix}\begin{bmatrix}\overset{\sim}{u} \\\overset{\sim}{v} \\\overset{\sim}{z}\end{bmatrix}} + \begin{bmatrix}{g_{u}\left( {\overset{\sim}{u},t} \right)} \\0 \\{g_{z}\left( {\overset{\sim}{z},t} \right)}\end{bmatrix}}\overset{.}{=}{{A_{w}\left( {\overset{\sim}{u},\overset{\sim}{v},\overset{\sim}{z}} \right)}^{T} + {g_{uz}\left( {\overset{\sim}{u},\overset{\sim}{z},t} \right)}}}}{{{where}\mspace{14mu}{g_{uz}\left( {\overset{\sim}{u},\overset{\sim}{z},t} \right)}} = {\left( {g_{u},0,g_{z}} \right)^{T}\mspace{14mu}{and}}}} & (47) \\{g_{z} - {\left( {1 + a - {3{z_{2}\left( {t - t_{d}} \right)}}} \right){\overset{\sim}{z}}^{2}} + {\left( {{2\left( {1 + a} \right){z_{2}\left( {t - t_{d}} \right)}} - {3{z_{2}^{2}\left( {t - t_{d}} \right)}}} \right)\overset{\sim}{z}} - {{\overset{\sim}{z}}^{3}.}} & (48)\end{matrix}$

Apparently if the origin (ũ, {tilde over (v)}, {tilde over (z)})=0 ofthe zero dynamics is asymptotically stable, then {tilde over (x)}converges to zero as {tilde over (w)} tends to zero.

In Equation (47), the matrix A_(w) is Hurwitz and the periodic signalsu₂(t−t_(d)) and z₂(t−t_(d)) are functions of the state x_(e) of theexosystem. In this case, in view of the functions g_(u) and g_(z) inEquation (47), one finds that the center manifold is (ũ,{tilde over(v)},{tilde over (z)})=(Ũ,{tilde over (V)},{tilde over (Z)})=0. Similarto the arguments used on either the Jacobian linearization or the centermanifold theorem, it can be concluded that for small(u₂(t−t_(d)),z₂(t−t_(d))), the origin of the zero dynamics isexponentially stable (in a local sense), and thereby localsynchronization is accomplished.

Simulation results are now presented for the closed-loop system ofEquation (5) and Equation (45). The parameter values, command inputsequence, and the initial conditions of FIG. 1(A)-(D) are retained. Thefeedback gain chosen is p₀=8. The responses are shown in FIG. 8(A)-(D)and FIG. 9(A)-(D). It is observed that synchronization has beenaccomplished following each change in the phase command signal, butconvergence time is larger. The plots of u₁ show high frequencyoscillation at certain instances, but it has not caused any problems.Only a small control magnitude has been used.

Simulation results are obtained for a different value of the parametera=0.01 and the time scaling factor is set to 100 giving the frequency ofoscillation close to one Hz. The closed-loop control system using eachof the control laws C_(u), C_(v) and C_(z) and C_(w) is simulated. Thecommand input, the feedback gains, and initial conditions of FIG.1(A)-(D) are retained for simulation. Results are presented only for theclosed-loop system including the simplest control law C_(w). Theresponses are shown in FIG. 10(A)-(D) through FIG. 12(A)-(D).

It is of interest to discuss the relative merits of the fourcontrollers. As indicated earlier, the first controller has a globalstabilization property and for the remaining controllers only localsynchronization has been established. It is important to note that onlya finite region of stability in the {tilde over (x)}-space existsbecause the local stability of the closed-loop system including thecontrollers C_(v), C_(z), and C_(w) has been proven. But it is expectedthat as the complexity of control law increases, the region of stabilityenlarges. For this reason, one expects that the control law C_(w) hasbeen proven. But it is expected that as the complexity of control lawincreases, the region of stability enlarges. For this reason, oneexpects that the control law C_(w) can accomplish synchronization onlyfor relatively small perturbations in {tilde over (x)} at the instantwhen the phase command is given. Of course, the error {tilde over (x)},and therefore the synchronization of the IOs, depends on the instant ofcontroller switching. Based on the simulation results, it has been foundthat the controllers C_(v) and C_(z) have fairly large regions ofstability and one does not necessarily have to use the controller C_(u),which has the highest degree of complexity among the derivedcontrollers. Unlike the global controller, the controllers C_(v), C_(z),and C_(w) provide smoother (z,w)-responses. This is due to thefast-varying nonlinear function of large magnitude in the control lawC_(u). It may be pointed out that there exists flexibility in thedesign, and by a proper choice of feedback gains and the reference phasecommand signals, one can obtain different response characteristics. Thisflexibility in phase control of IOs is useful in performing desirablemaneuvers of the BAUV.

In the derivation of the control laws, it is assumed that the IOs areidentical. While for the BAUV application, it is appropriate to havesimilar parameters, it is pointed out that the design approach is quitegeneral, and it is applicable to nonidentical IOs having differentparameters. The design has been presented only for two IOs, but it isstraightforward to extend the derivation for the synchronization of anynumber of IOs.

Advantages and Disadvantages

The IOs have complex nonlinear dynamics. As such, controllers (PID,optimal, lead-lag compensation, etc.) designed using linearized modelscannot guarantee global synchronization. One must note that the profileof the control signal will depend on the states of the IOs when thepulse is applied. The derived controllers are based on the input-outputfeedback linearization theory, and stability and convergence. Thedesigned global controller accomplishes synchronization for all initialconditions. Moreover, design parameters provide flexibility in shapingresponse characteristics. The controller can be switched on for phasecontrol at any instant since the controller utilizes state variablefeedback and one can command the IO to follow a sequence of phasechanged when needed for the control of the BAUV. This is especiallyimportant if operating fins of the BAUV operate at low frequencies. Thecontrol laws are explicit functions of the state variables of the IOsand can be easily implemented.

The foregoing description of the preferred embodiments of the inventionhas been presented for purposes of illustration and description only. Itis not intended to be exhaustive nor to limit the invention to theprecise form disclosed; and obviously many modifications and variationsare possible in light of the above teaching. Such modifications andvariations that may be apparent to a person skilled in the art areintended to be included within the scope of this invention as defined bythe accompanying claims.

1. A control system for maneuvering an underwater vehicle, said controlsystem comprising: a propulsor system positioned on the underwatervehicle; and a controller operationally connected to said propulsorwherein said controller is capable recognizing at least two inferiorolives wherein a first inferior olive of the inferior olives oscillatesin synchronism with a predetermined delay time t and a phase anglecorresponding to a second inferior olive of the inferior olives toresolve nonlinear functions in response to disturbances whenmaneuvering; wherein the inferior olives are controlled bysynchronization of initial conditions of the first inferior olive andthe second inferior olive wherein a controlled output variable is chosenase(t)=h _(u)(x ₁(t), x ₂(t−t _(d)))=u ₁(t)−u₂(t−t _(d)) wherein acomposite state vector for the inferior olives is defined asx_(a)(t)=(x₁(t)^(T), x₂(t−t_(d))^(T) ε R⁸ and a vector field is definedby${{L_{f}{h_{u}\left( {x_{a}(t)} \right)}} = {{\frac{\partial h}{\partial x_{a}}{f\left( {x_{a}(t)} \right)}} = {{\frac{\partial h_{u}}{\partial x_{1}}{f_{1}\left( {x_{1}(t)} \right)}} + {\frac{\partial h_{u}}{\partial x_{2}}{f_{2}\left( {x_{2}\left( {t - t_{d}} \right)} \right)}}}}};$resolving  L_(f)^(i)h_(u)(x_(a)) = L_(f)L_(f)^(j − 1)h_(u)(x_(a)(t))  and${L_{g}L_{f}^{k}{h_{u}\left( x_{a} \right)}} = {\frac{{\partial L_{f}^{k}}h_{u}}{\partial x_{a}}g}$wherein an input-output linearizing control law for the inferior olivesprogrammable to the controller is selected by$u_{c\; 1} = {{b_{u\; 1}^{- 1}\text{(}} - a_{u\; 1} - {\sum\limits_{j = 0}^{3}{p_{j}L_{f}^{j}{{h_{u}\left( {x_{a}(t)} \right)}.}}}}$2. A method for maneuvering an underwater vehicle, said methodcomprising the steps of: providing at least two inferior olives;resolving e=h(x₁(t), x₂(t−t_(d))); choosing an output variablee(t)=h _(u)(x ₁(t), x ₂(t−t _(d)))=u ₁(t)−u ₂(t−t _(d)); defining acomposite state vector for the inferior olives asx _(a)(t)=(x ₁(t)^(T) , x ₂(t−t _(d))^(T) ε R ⁸; defining along a vectorfield${{L_{f}{h_{u}\left( {x_{a}(t)} \right)}} = {{\frac{\partial h}{\partial x_{a}}{f\left( {x_{a}(t)} \right)}} = {{\frac{\partial h_{u}}{\partial x_{1}}{f_{1}\left( {x_{1}(t)} \right)}} + {\frac{\partial h_{u}}{\partial x_{2}}{f_{2}\left( {x_{2}\left( {t - t_{d}} \right)} \right)}}}}};$resolving  L_(f)^(i)h_(u)(x_(a)) = L_(f)L_(f)^(j − 1)h_(u)(x_(a)(t))  and${{L_{g}L_{f}^{k}{h_{u}\left( x_{a} \right)}} = {\frac{{\partial L_{f}^{k}}h_{u}}{\partial x_{a}}g}};$selecting an input-output linearizing control law${{{by}\mspace{14mu} u_{c\; 1}} = {{b_{u\; 1}^{- 1}\text{(}} - a_{u\; 1} - {\sum\limits_{j = 0}^{3}{p_{j}L_{f}^{j}{h_{u}\left( {x_{a}(t)} \right)}}}}};$producing an output equation of the forme ⁽⁴⁾ +p ₃ e ⁽³⁾ +p ₂ e ⁽²⁾ +p ₁ ė+p ₀ e=0; synchronizing the inferiorolives wherein a first inferior olive of the inferior olives oscillatesin synchronism with a delay time corresponding to a desired phase anglewith respect to a second inferior olive of the inferior olives;processing the synchronized inferior olives with a controller; andmaneuvering a propulsor of the underwater vehicle with the controller.3. The method in accordance with claim 2, further comprising the step ofobtaining frequencies of the inferior olives by time scaling.
 4. Amethod for controlling an underwater vehicle, said method comprising thesteps of: providing at least two inferior olives; choosing an outputvariable for the inferior olivese(t)=h _(v)(x _(a)(t))=v ₁(t)−v ₂(t−t _(d))={tilde over (v)}(t);selecting an input-output linearizing control law by${u_{c\; 1} = {b_{v\; 1}^{- 1}\left( {{- a_{v\; 1}} - {\sum\limits_{j = 0}^{2}{p_{j}L_{f}^{j}{h_{v}\left( {x_{a}(t)} \right)}}}} \right)}};$determining an output equation e⁽³⁾+p₂e⁽²⁾+p₁ė+p₀e=0; choosing gainsp_(i) such that a characteristic polynomial isΠ_(v)(λ)=λ³ +p ₂λ² +p ₁ λ+p ₀; establishing residual dynamics such thatan equilibrium point is asymptotically stable; achieving localsynchronization of the inferior olives wherein a first inferior olive ofthe inferior olives oscillates in synchronism with a delay timecorresponding to a desired phase angle with respect to a second inferiorolive of the inferior olives in a closed system; processing thesynchronized inferior olives with a controller; and maneuvering apropulsor of the underwater vehicle with the controller.
 5. The methodin accordance with claim 4, said method further comprising the step ofestablishing asymptotic stability of the zero dynamics using a centermanifold theorem.
 6. A method for controlling an underwater vehicle,said method comprising the steps of: providing at least two inferiorolives; choosing an output variable e(t)=z₁(t)−z₂(t−t_(d))=h_(z)(x_(a));selecting an input-output linearizing control law by${u_{c\; 1} = {{b_{z\; 1}^{- 1}\text{(}} - a_{z\; 1} - {\sum\limits_{j = 0}^{1}{p_{j}L_{f}^{j}{h_{v}\left( {x_{a}(t)} \right)}}}}};$determining an output equation e⁽²⁾+p₁ė+p₀e=0; choosing gains p_(i) suchthat a characteristic polynomial isΠ_(z)(λ)=λ² +p ₁ λ+p ₀; defining a composite state vector for theinferior olives asx _(a)(t)=(x ₁(t)^(T) , x ₂(t−t _(d))^(T) ε R ⁸; establishing residualdynamics wherein an equilibrium point is asymptotically stable;achieving local synchronization of the inferior olives wherein a firstinferior olive of the inferior olives oscillates in synchronism with adelay time corresponding to a desired phase angle with respect to asecond inferior olive of the inferior olives in a closed system;processing the synchronized inferior olives with a controller; andmaneuvering a propulsor of the underwater vehicle with the controller.7. A method for controlling an underwater vehicle, said methodcomprising the steps of: providing at least two inferior olives;choosing an output variablee(t)=w ₁(t)−w ₂(t−t _(d))={tilde over (w)}=h _(w)(x _(a)(t)); selectingan input-output control law by u_(c1)={tilde over (z)}(t)+p₀ ε_(Ca) ⁻¹{tilde over (w)} thereby satisfying an output with {tilde over ({dotover (w)}+p₀{tilde over (w)}=0 and in a closed-loop system {tilde over(w)} tends to zero; establishing residual dynamics wherein anequilibrium point is asymptotically stable; achieving localsynchronization of the inferior olives wherein a first inferior olive ofthe inferior olives oscillates in synchronism with a delay timecorresponding to a desired phase angle with respect to a second inferiorolive of the inferior olives in the closed system; processing thesynchronized inferior olives with a controller; and maneuvering apropulsor of the underwater vehicle with the controller.